If the car radiator is empty have no water will the heat gauge in the car show high temp?

No, as there will be no liquid near the sensor.

You will burn your engine up.

If the coolant can't reach the temp sensor then no.

It will only take a few minutes running to overheat an engine with an empty radiator. The gauge will show such by conduction. of heat through its all metal body from the very hot cylinder head and block, which is what heats coolant. If you want proof, stick your hand on a cylinder head after the engine has run only 5 minutes.

No. The motor will overheat way before the temperature gauge gets hot. There has to be coolant touching the gauge sending unit to get an accurate reading.

Yes, especially right before the engine seizes.

Nope - must be immersed in the coolant. Back when they made English cars, they were experts at putting the sensors in areas that would be the first to run dry . . . and topped it all with the TR7 where the top of the engine was way higher than the radiator so it was even difficult to fill the cooling system completely !!

If the engine uses a coolant temp sensor then it most likely will not reflect that the engine is running super hot, at least not until it's way too late, if the engine has no coolant in it. If the coolant temp sensor isn't immersed in coolant, it can't read a proper temp.

If the engine uses a block temperature sensor instead of a coolant temp sensor to read engine temp, then it should still read correctly (HOT!) even if there is no coolant whatsoever because that type of sensor doesn't use the coolant to determine engine temp in the first place. Without knowing anything about your actual car I can't say which type of temp sensor it uses.

Yes. Just before you blow the head gasket.

Peace.

No. It probably will not.